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Mirrors > Home > MPE Home > Th. List > 0e0iccpnf | Structured version Visualization version GIF version |
Description: 0 is a member of (0[,]+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0e0iccpnf | ⊢ 0 ∈ (0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 9965 | . 2 ⊢ 0 ∈ ℝ* | |
2 | 0le0 10987 | . 2 ⊢ 0 ≤ 0 | |
3 | elxrge0 12152 | . 2 ⊢ (0 ∈ (0[,]+∞) ↔ (0 ∈ ℝ* ∧ 0 ≤ 0)) | |
4 | 1, 2, 3 | mpbir2an 957 | 1 ⊢ 0 ∈ (0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-icc 12053 |
This theorem is referenced by: xrge0subm 19606 itg2const2 23314 itg2splitlem 23321 itg2split 23322 itg2gt0 23333 itg2cnlem2 23335 itg2cn 23336 iblss 23377 itgle 23382 itgeqa 23386 ibladdlem 23392 iblabs 23401 iblabsr 23402 iblmulc2 23403 bddmulibl 23411 xrge0infss 28915 xrge00 29017 unitssxrge0 29274 xrge0mulc1cn 29315 esum0 29438 esumpad 29444 esumpad2 29445 esumrnmpt2 29457 esumpinfval 29462 esummulc1 29470 ddemeas 29626 oms0 29686 itg2gt0cn 32635 ibladdnclem 32636 iblabsnc 32644 iblmulc2nc 32645 bddiblnc 32650 ftc1anclem7 32661 ftc1anclem8 32662 ftc1anc 32663 iblsplit 38858 gsumge0cl 39264 sge0cl 39274 sge0ss 39305 0ome 39419 ovnf 39453 |
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